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Section 4.2 What Derivatives Tell UsTopic 1: Increasing and Decreasing FunctionsDefinition Increasing and Decreasing FunctionsSuppose a function f is defined on an interval I. We say that f is increasing on I if fx2>f(x1) whenever x1 and x2 are in I and x2>x1. We say that f is decreasing on I if fx2<f(x1) whenever x1 and x2 are in I and x2>x1. Topic 2: Intervals of Increase and DecreaseRecall that the derivative of a function gives the slope of the tangent lines. If the derivative is positive on an interval, the tangent lines on that interval have positive slopes, and the function is increasing. If the derivative is negative on an interval, the tangent lines on that interval have negative slopes, and the function is decreasing. TheoremTest for Intervals of Increase and DecreaseSuppose f is continuous on an interval I and differentiable at all interior points of I. If f'x>0 at all interior points of I, then f is increasing on I. If f'x<0 at all interior points of I, then f is decreasing on ic 3: Identifying Local Maxima and MinimaSuppose x=c is a critical point of f, where f'c=0.Suppose also that f' changes signs at c with f'x<0 on an interval (a,c) to the left of c and f'x>0 on an interval (c,b) to the right of c. In this case, f is decreasing to the left of c and increasing to the right of c. Thus, f has a local minimum at c.Similarly, suppose that f' changes signs at c with f'x>0 on an interval (a,c) to the left of c and f'x<0 on an interval (c,b) to the right of c. In this case, f is increasing to the left of c and decreasing to the right of c. Thus, f has a local maximum at c.TheoremFirst Derivative Test Suppose that f is continuous on an interval that contains a critical point c and assume f is differentiable on an interval containing c, except perhaps at c itself.If f' changes sign from positive to negative as x increases through c, then f has a local maximum at c.If f' changes sign from negative to positive as x increases through c, then f has a local minimum at c.If f' is positive on both sides near c or negative on both sides near c, then f has no local extreme value at ic 4: Concavity and Inflection Points right35433000Consider the function fx=x3. Its graph bends upward for x>0, reflecting the fact that the tangent lines get steeper as x increases. It follows that the first derivative is increasing for x>0. A function with the property that f' is increasing on an interval is concave up on that interval.Similarly, the graph of fx=x3 bends downward for x<0 because it has a decreasing first derivative on that interval. A function with the property that f' is decreasing on an interval is concave down on that interval. If a function is concave up at a point, then the tangent line will lie below the graph of the function. If a function is concave down at a point, then the tangent line will lie above the graph of the function. Definition Concavity and Inflection Points Let f be differentiable on an open interval I. If f' is increasing on I, then f is concave up on I. If f' is decreasing on I, then f is concave down on I. If f is continuous at c and f changes concavity at c (from up to down or down to up), then f has an inflection point at c. If f″ is positive on an interval I, then f' is increasing on I, and f is concave up on I. If f″ is negative on an interval I, then f' is decreasing on I, and f is concave down on I. If the values of f″ change signs at a point c, then the concavity of f changes at c, and f has an inflection point at c. TheoremTest for ConcavitySuppose that f″ exists on an open interval I.If f″ is positive on I, then f is concave up on I.If f″ is negative on I, then f is concave down on I.If c is a point of I and f″ changes sign at c, then f has an inflection point at c. If f″c=0 or f″(c) does not exist, then (c,fc) is a candidate for an inflection point. To determine whether or not an inflection point occurs at c, it is necessary to determine if the concavity of f changes at ic 5: Second Derivative TestTheoremSecond Derivative Test for Local ExtremaSuppose that f″ is continuous on an open interval containing c with f'c=0.If f″c>0, then f has a local minimum at c.If f″c<0, then f has a local maximum at c.If f″c=0, then the test is inconclusive; f may have a local maximum, local minimum, or neither at c. Topic 6: A Summary of Derivative Properties ................
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